C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 218–224 (2017) D O I: 10.1501/C om mua1_ 0000000813 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
-RICCI SOLITONS IN TRANS-SASAKIAN MANIFOLDS
K. VENU AND H. G. NAGARAJA
Abstract. The aim of this paper is to study the -Ricci solitons in 3-dimensional trans-Sasakian manifolds.
1. Introduction
A class of almost contact metric manifolds known as trans-Sasakian manifolds was introduced by Oubino[11] in 1985. In [4], Gray-Hervella classi…cation of almost Hermite manifolds appears as a class W4 of Hermitian manifolds which are closely
related to locally conformally Kähler manifolds. An almost contact metric structure on a manifold M is called a trans-Sasakian structure if the product manifold M R belongs to the class W4. The trans-Sasakian structures also provide a large class
of generalized quasi-Sasakian structures. The local structures of trans-Sasakian manifolds of dimension n 5 has been completely characterized by Marrero [7]. He proved that a trans-Sasakian manifold of dimension n 5 is either cosymplectic or Sasakian or Kenmotsu manifold.
In 1982, Hamilton [5] made the fundamental observation that Ricci ‡ow is an excellent tool for simplifying the structure of a manifold. It is a process which deforms the metric of a Riemannian manifold by smoothing out the irregularities. It is given by
@g
@t = 2Ric g: (1.1)
Ricci soliton in a Riemannian manifold (M; g) is a special solution to the Ricci ‡ow and is a natural generalization of an Einstein metric which is de…ned as a triple (g; V; ) with g a Riemannian metric, V a vector …eld and a real scalar such that LVg(X; Y ) + 2S(X; Y ) + 2 g(X; Y ) = 0; (1.2)
where S is the Ricci tensor of M and LV denote the Lie derivative operator along
the vector …eld V .
Received by the editors: May 03, 2016; Accepted: February 02, 2017. 2010 Mathematics Subject Classi…cation. 53C20, 53C44.
Key words and phrases. -Ricci soliton, quasi conformal curvature tensor, pseudo-projective curvature tensor, trans-Sasakian manifold.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
The Ricci soliton is said to be shrinking, steady and expanding accordingly as is negative, zero and positive respectively. In [13], Sharma initiated the study of Ricci solitons in contact Riemannian geometry. Later Tripathi [14], Nagaraja et al.[10] and others extensively studied Ricci solitons in contact metric manifolds.
It is well known that, if the potential vector …eld is zero or Killing then the Ricci soliton is an Einstein metric. In [6], [2] and [9], the authors proved that there are no Einstein real hypersurfaces of non-‡at complex space forms. Moti-vated by this the authors Cho and Kimura [3] introduced the notion of - Ricci solitons and gave a classi…cation of real hypersurfaces in non-‡at complex space forms admitting - Ricci solitons. Later Blaga [1] studied -Ricci solitons in para-Kenmotsu manifolds. Recently, Prakasha and Hadimani [12] studied -Ricci soli-tons on para-Sasakian manifolds. It is quite interesting to study - Ricci solisoli-tons in trans-Sasakian manifolds not as real hypersurfaces of complex space forms but as special contact structures. In this paper, we derive the condition for a 3 dimen-sional trans-Sasakian manifold as an - Ricci soliton and derive expression for the scalar curvature.
2. Preliminaries
A di¤erentiable manifold M is said to be an almost contact metric manifold if it admits a (1; 1) tensor …eld , a vector …eld , a 1-form and Riemannian metric g, which satisfy
2
= I + ; ( ) = 1; = 0; = 0; (2.1)
g( X; Y ) = g(X; Y ) (X) (Y ); g(X; ) = (X); (2.2) for all vector …elds X; Y on M .
An almost contact metric manifold M ( ; ; ; g) is said to be trans-Sasakian manifold, if (M R; J; G) belongs to class W4 of the Hermitian manifold, where J
is the almost complex structure of M R de…ned by
J (Z; f d=dt) = ( Z f ; (Z)d=dt); (2.3)
for all vector …elds Z on M and smooth function f on M R and G is the product metric on M R. This is expressed by the following condition
(rX )Y = (g(X; Y ) (Y )X) + (g( X; Y ) (Y ) X); (2.4)
where and are some scalar functions and such a structure is said to be the trans-Sasakian structure of type ( ; ). We note that trans-trans-Sasakian manifolds of type (0; 0); ( ; 0) and (0; ) are the cosymplectic, -Sasakian and -Kenmotsu manifold respectively. In particular, if = 1; = 0 and = 0; = 1, then a trans-Sasakian manifold reduces to a Sasakian and Kenmotsu manifolds respectively. From (2.4), it follows that
rX = X + (X (X) ); (2.5)
and
The trans-Sasakian manifold with structure tensor ( ; ; ; g) on M satis…es the following relations: R(X; Y ) =( 2 2)[ (Y )X (X)Y ] + 2 [ (Y ) X (X) Y ] + (Y ) X (X ) Y + (Y ) 2X (X ) 2Y; 2 + = 0; (2.7) S(X; ) = (n 1)( 2 2) ) (X) (n 2)(X ) ( X) ; Q = (n 1)( 2 2) ) (n 1)grad + (grad ); (2.8)
where R is curvature tensor, while Q is the Ricci operator given by S(X; Y ) = g(QX; Y ).
Further in a trans-Sasakian manifold of type ( ; ), we have
(grad ) = (n 1)grad : (2.9)
Using (2.7) and (2.9), for constants and , we have
R( ; X)Y = ( 2 2)[g(X; Y ) (Y )X] (2.10)
R(X; Y ) = ( 2 2)[ (Y )X (X)Y ] (2.11)
(R(X; Y )Z) = ( 2 2)[g(Y; Z) (X) g(X; Z) (Y )] (2.12)
S(X; ) = (n 1)( 2 2) (X) (2.13)
Q = (n 1)( 2 2) : (2.14)
An important consequence of (2.5) is that is a geodesic vector …eld.
i.e., r = 0: (2.15)
For arbitrary vector …eld X, we have that d ( ; X) = 0:
The -sectional curvature K of (M; g) is the sectional curvature of a plane spanned by and a unitary vector …eld X. From (2.11), we have
K = g(R(X; ) ; X) = ( 2 2): (2.16)
It follows from (2.16) that sectional curvature does not depend on X. 3. -Ricci solitons on (M; ; ; ; g)
Consider now a symmetric tensor …eld h of (0; 2) - type which is parallel with respect to Levi-Civita connection (rh = 0). Applying the Ricci commutation identity r2h(X; Y ; Z; W ) r2h(X; Y ; W; Z) = 0, we obtain
Replacing Z = W = in (3.1) and by the symmetry of h, we have
h(R(X; Y ) ; ) = 0: (3.2)
Taking X = in (3.2), then by virtue of (2.11), we have
( 2 2)[h(Y; ) h( ; ) (Y ) = 0: (3.3)
With the hypothesis on K , the above equation yields:
h(Y; ) = h( ; )g(Y; ): (3.4)
Again by taking X = Z = in (3.1), we obtain
( 2 2)[ (Y )h( ; W ) h(Y; W ) + g(Y; W )h( ; ) (W )h( ; Y )] = 0: (3.5) Since ( 2 2) 6= 0, we have
h(Y; W ) = (Y )h( ; W ) + g(Y; W )h( ; ) (W )h( ; Y ): (3.6) By using (3.4) in (3.6), we get
h(X; Y ) = h( ; )g(X; Y ): (3.7)
The above equation gives the conclusion :
Theorem 3.1. Let (M; ; ; ; g) be a trans-Sasakian manifold with non-vanishing -sectional curvature and endowed with a tensor …eld h 2 (T20(M )) which is
sym-metric and -skew-symsym-metric. If h is parallel with respect to r then it is a constant multiple of the metric tensor g.
We call the notion of -Ricci solitons from [3].
LVg(X; Y ) + 2S(X; Y ) + 2 g(X; Y ) + 2 (X) (Y ) = 0; (3.8)
where LV is the Lie derivative operator along the vector …eld V and and are
real constants.
Because of (2.5), the equation (3.8) becomes:
S(X; Y ) = ( + )g(X; Y ) + ( ) (X) (Y ): (3.9)
The above equation yields
S(X; ) = ( + ) (X); (3.10)
QX = ( + )X + ( ) (X) ; (3.11)
Q = ( + ) ; (3.12)
r = n (n 1) ; (3.13)
where r is the scalar curvature. O¤ the two natural situations regarding the vector …eld V : V 2 Span and V ? , we investigate only the case V = .
Our interest is in the expression for L g + 2S + 2 . A straightforward computation gives
In a 3-dimensional trans-Sasakian manifold, we have
R(X; Y )Z =g(Y; Z)QX g(X; Z)QY + S(Y; Z)X S(X; Z)Y r 2[g(Y; Z)X g(X; Z)Y ]; (3.15) QX =h r 2 + ( ) ( 2 2 ) i X h r 2+ ( ) 3( 2 2 ) i (X) + (X)[ (grad ) grad ] [(( X) ) + (X )] : (3.16) By using (2.7) and (2.9) in (3.16), we obtain
QX =h r 2 ( 2 2 ) i X h r 2 3( 2 2 ) i (X) ; (3.17) S(X; Y ) =h r 2 ( 2 2)ig(X; Y ) h r 2 3( 2 2)i (X) (Y ): (3.18)
Next, we consider the equation
h(X; Y ) = (L g)(X; Y ) + 2S(X; Y ) + 2 (X) (Y ): (3.19) By using (3.14) and (3.18) in (3.19), we have
h(X; Y ) =h r 2 2( 2 2 ) + 2 i g(X; Y ) h r 2 6( 2 2) + 2 + 2 i (X) (Y ): (3.20) Putting X = Y = in (3.20), we get h( ; ) = 2[2( 2 2) ]: (3.21) So (3.7) becomes h(X; Y ) = 2[2( 2 2) ]g(X; Y ): (3.22)
From (3.19) and (3.22), it follows that g is an -Ricci soliton. So we can state:
Theorem 3.2. Let (M3; ; ; ; g) be a 3-dimensional trans-Sasakian manifold.
Then (g; ; ) yields an -Ricci soliton on (M3; ; ; ; g).
Let V be pointwise collinear with . i.e., V = b , where b is a function on the 3-dimensional trans-Sasakian manifold. Then
g(rXb ; Y ) + g(rYb ; X) + 2S(X; Y ) + 2 g(X; Y ) + 2 (X) (Y ) = 0; or bg((rX ; Y ) + (Xb) (Y ) + bg(rY ; X) + (Y b) (X) + 2S(X; Y ) + 2 g(X; Y ) + 2 (X) (Y ) = 0: Using (2.5), we obtain bg( X + (X (X) ); Y ) + (Xb) (Y ) + bg( Y + (Y (Y ) ); X) + (Y b) (X) + 2S(X; Y ) + 2 g(X; Y ) + 2 (X) (Y ) = 0;
which yields
2b g(X; Y ) 2b (X) (Y ) + (Xb) (Y ) + (Y b) (X) + 2S(X; Y )
+ 2 g(X; Y ) + 2 (X) (Y ) = 0: (3.23)
Replacing Y by in the above equation, we obtain
Xb + ( b) (X) + 2(2( 2 2) + + ) (X) = 0: (3.24)
Again putting X = in (3.24), we obtain b = 2( 2 2) :
Plugging this in (3.24), we get Xb + +(2( 2 2 ) + + ) (X) = 0, or db = f + + 2( 2 2)g : (3.25) Applying d on (3.25), we get f + + 2( 2 2 )gd = 0: Since d 6= 0 we have + + 2( 2 2) = 0: (3.26)
Equation (3.26) in (3.25) yields b as a constant. Therefore from (3.23), it follows that
S(X; Y ) = ( + b )g(X; Y ) + (b ) (X) (Y );
which implies that M is of constant scalar curvature for a constant . This leads to the following:
Theorem 3.3. If in a 3-dimensional trans-Sasakian manifold the metric g is an -Ricci soliton and V is pointwise collinear with , then V is a constant multiple of and g is of constant scalar curvature provided is a constant.
Taking X = Y = in (3.7) and (3.18) and comparing, we get
= 2( 2 2) + = 2K : (3.27)
From (3.13) and (3.27), we obtain
r = 6( 2 2) 2 + 2 : (3.28)
Since is a constan,t it follows from (3.27) that K is a constant.
Theorem 3.4. Let (g; ; ) be an -Ricci soliton in (M3; ; ; ; g). Then the scalar
and the scalar curvature r satisfy the relations:
+ = 2K , r = 6K + 2 + 2 .
Remark 3.5: For = 0, (3.27) reduces to = 2K , so Ricci soliton in 3-dimensional trans-Sasakian manifold is shrinking.
References
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Current address : Venu K and H. G. Nagaraja: Department of Mathematics, Bangalore Uni-versity, Bengaluru-560056, INDIA
E-mail address : venuk.math@gmail.com E-mail address : hgnraj@yahoo.com
